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Section 5.1 Degree of Freedom

Degrees of freedom refers to the number of independent parameters or values required to specify the state of an object.

The state of a particle is completely specified by its location in space, while the state of a rigid body includes its location in space and also its orientation.

Two-dimensional rigid bodies in the \(xy\) plane have three degrees of freedom. Position can be characterized by the \(x\) and \(y\) coordinates of a point on the object, and orientation by angle \(\theta_z\) about an axis perpendicular to the plane. The complete movement of the body can be defined by two linear displacements \(\Delta x\) and \(\Delta y\text{,}\) and one angular displacement \(\Delta \theta_z\text{.}\)

Figure 5.1.1. Two-dimensional rigid bodies have three degrees of freedom.

Three-dimensional rigid bodies have six degrees of freedom, which can be specified with three orthogonal coordinates \(x, y\) and \(z\text{,}\) and three angles of rotation, \(\theta_x, \theta_y\) and \(\theta_z\text{.}\) Movement of the body is defined by three translations \(\Delta x\text{,}\) \(\Delta y\) and \(\Delta z\text{,}\) and three rotations \(\Delta \theta_x\text{,}\) \(\Delta \theta_y\) and \(\theta_z\text{.}\)

Figure 5.1.2. Three-dimensional rigid body have six degrees of freedom - three translations and three rotations.

For a body to be in static equilibrium, all possible movements of the body need to be adequately restrained. If a degree of freedom is not restrained, the body is in an unstable state, free to move in one or more ways. Stability is highly desirable for reasons of human safety, and bodies are often restrained by redundant restraints so that if one were to fail, the body would still remain stable. If the restraints correctly interpreted, then equal constraints and degrees of freedom create a stable system, and the values of the reaction forces and moments can be determined using equilibrium equations. If the number of restraints exceeds the number of degrees of freedom, the body is in equilibrium but you will need techniques we won't cover in statics to determine the reactions.